Generic properties concerning well-posed optimization problems. (English) Zbl 0576.49018
Summary: We consider a class of (constrained) minimization problems: find \(x_ 0\in A\) such that \(f(x_ 0)=\min\) \(\{\) f(x): \(x\in A\}\). Here A is an arbitrary non-empty closed subset of a complete metric space X and \(f: X\to R\) is an arbitrary continuous real-valued function bounded from below. Obviously, every problem of this class is determined by the pair (A,f). The set of all these pairs is endowed with a natural complete metric.
Our aim is to show that the ’majority’ of the problems from the above class are well-posed (i.e. there is uniqueness and continuous dependence of solutions on A and f; the well-posedness in this article means that both Hadamard and Tykhonov well-posedness are fulfilled). Here the ’majority’ is understood in the Baire category sense, viz. the set of well-posed problems contains a dense \(G_{\delta}\)-subset of all the complete metric space, i.e. its complement is of first Baire category and is considered to be a small set. Several classes of convex optimization problems are investigated to be a small set. Several classes of convex optimization problems are investigated in the same direction.
Our aim is to show that the ’majority’ of the problems from the above class are well-posed (i.e. there is uniqueness and continuous dependence of solutions on A and f; the well-posedness in this article means that both Hadamard and Tykhonov well-posedness are fulfilled). Here the ’majority’ is understood in the Baire category sense, viz. the set of well-posed problems contains a dense \(G_{\delta}\)-subset of all the complete metric space, i.e. its complement is of first Baire category and is considered to be a small set. Several classes of convex optimization problems are investigated to be a small set. Several classes of convex optimization problems are investigated in the same direction.
MSC:
49K40 | Sensitivity, stability, well-posedness |
49K27 | Optimality conditions for problems in abstract spaces |
90C48 | Programming in abstract spaces |
46A55 | Convex sets in topological linear spaces; Choquet theory |
90C25 | Convex programming |